- Pre-Abelian category
In

mathematics , specifically incategory theory , a**pre-Abelian category**is anadditive category that has all kernels and cokernels.Spelled out in more detail, this means that a category

**C**is pre-Abelian if:

#**C**is preadditive, that is enriched over themonoidal category ofAbelian group s;

#**C**has allbiproduct s, which are both finite products and finitecoproduct s;

# given any morphism "f": "A" → "B" in**C**, theequaliser of "f" and thezero morphism from "A" to "B" exists (this is the kernel), as does thecoequaliser (this is the cokernel).Note that the zero morphism in item 3 can be identified as theidentity element of thehom-set Hom("A","B"), which is an Abelian group by item 1; or as the unique morphism "A" → "O" → "B", where "O" is azero object , guaranteed to exist by item 2.**Examples**The original example of an additive category is the category

**Ab**ofAbelian group s.**Ab**is preadditive because it is aclosed monoidal category , the biproduct in**Ab**is the finitedirect sum , the kernel is inclusion of the ordinary kernel from group theory and the cokernel is the quotient map onto the ordinary cokernel from group theory.Other common examples:

* The category of (left) modules over a ring "R", in particular:

** the category ofvector space s over a field "K".

* The category of (Hausdorff) abeliantopological group s.These will give you an idea of what to think of; for more examples, seeAbelian category (every Abelian category is pre-Abelian).**Elementary properties**Every pre-Abelian category is of course an

additive category , and many basic properties of these categories are described under that subject.This article concerns itself with the properties that exist specifically because of the existence of kernels and cokernels.Although kernels and cokernels are special kinds of

equaliser s andcoequaliser s, a pre-Abelian category actually has "all" equalisers and coequalisers.We simply construct the equaliser of two morphisms "f" and "g" as the kernel of their difference "g" − "f"; similarly, their coequaliser is the cokernel of their difference.(The alternative term "difference kernel" for binary equalisers derives from this fact.)Since pre-Abelian categories have all finite products andcoproduct s (the biproducts) and all binary equalisers and coequalisers (as just described), then by a general theorem ofcategory theory , they have all limits andcolimit s.That is, pre-Abelian categories are finitely complete.The existence of both kernels and cokernels gives a notion of image and

coimage .We can define these as:im "f" := ker coker "f";:coim "f" := coker ker "f".That is, the image is the kernel of the cokernel, and the coimage is the cokernel of the kernel.Note that this notion of image may not correspond to the usual notion of image, or range, of a function, even assuming that the morphisms in the category "are" functions.For example, in the category of topological Abelian groups, the image of a morphism actually corresponds to the inclusion of the "closure" of the range of the function.For this reason, people will often distinguish the meanings of the two terms in this context, using "image" for the abstract categorical concept and "range" for the elementary function-theoretic concept.

In many common situations, such as the category of sets, where images and coimages exist, their objects are

isomorphic .Put more precisely, we have a factorisation of "f": "A" → "B" as:"A" → "C" → "I" → "B",where the morphism on the left is the coimage, the morphism on the right is the image, and the morphism in the middle (called the "parallel" of "f") is an isomorphism.In a pre-Abelian category, "this is not necessarily true".The factorisation shown above does always exist, but the parallel might not be an isomorphism.In fact, the parallel of "f" is an isomorphism for every morphism "f"

if and only if the pre-Abelian category is anAbelian category .An example of a non-Abelian, pre-Abelian category is, once again, the category of topological Abelian groups.As remarked, the image is the inclusion of the "closure" of the range; however, the coimage is a quotient map onto the range itself.Thus, the parallel is the inclusion of the range into its closure, which is not an isomorphism unless the range was already closed.**Exact functors**Recall that all finite limits and

colimit s exist in a pre-Abelian category.In generalcategory theory , a functor is called "left exact" if it preserves all finite limits and "right exact" if it preserves all finite colimits. (A functor is simply "exact" if it's both left exact and right exact.)In a pre-Abelian category, exact functors can be described in particularly simple terms.First, recall that an

additive functor is a functor "F":**C**→**D**betweenpreadditive categories that acts as agroup homomorphism on eachhom-set .Then it turns out that a functor between pre-Abelian categories is left exactif and only if it is additive and preserves all kernels, and it's right exact if and only if it's additive and preserves all cokernels.Note that an exact functor, because it preserves both kernels and cokernels, preserves all images and coimages.Exact functors are most useful in the study of

Abelian categories , where they can be applied toexact sequence s.**Special cases*** An "

Abelian category " is a pre-Abelian category such that everymonomorphism andepimorphism is normal.The pre-Abelian categories most commonly studied are in fact Abelian categories; for example,**Ab**is an Abelian category.**References***

Nicolae Popescu ;1973 ; Abelian Categories with Applications to Rings and Modules; Academic Press, Inc.; out of print

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